Observation of Electron Coherence and FabryâPerot Standing Waves

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Observation of electron coherence and FabryPerot standing waves at a graphene edge Monica Allen, Oles Shtanko, Ion Cosma Fulga, Joel I-Jan Wang, Daniyar Nurgaliev, Kenji Watanabe, Takashi Taniguchi, Anton Akhmerov, Pablo Jarillo-Herrero, Leonid Levitov, and Amir Yacoby Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03156 • Publication Date (Web): 18 Oct 2017 Downloaded from http://pubs.acs.org on October 18, 2017

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Observation of electron coherence and Fabry-Perot standing waves at a graphene edge Monica T. Allen,∗,† Oles Shtanko,‡ Ion C. Fulga,¶,§ Joel I.-J. Wang,‡ Daniyar Nurgaliev,† Kenji Watanabe,k Takashi Taniguchi,k Anton R. Akhmerov,⊥ Pablo Jarillo-Herrero,‡ Leonid S. Levitov,‡ and Amir Yacoby∗,†,# †Department of Physics, Harvard University. 17 Oxford St. Cambridge, MA 02138. ‡Department of Physics, Massachusetts Institute of Technology. 77 Massachusetts Ave. Cambridge, MA 02139. ¶Department of Condensed Matter Physics, Weizmann Institute of Science. 234 Herzl Street, Rehovot 7610001 Israel. §Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany. kEnvironment and Energy Materials Division, National Institute for Materials Science. 1-1 Namiki, Tsukuba, Ibaraki, 305-0044 Japan. ⊥Kavli Institute of Nanoscience. Delft University of Technology. Lorentzweg 1. 2628 CJ Delft. The Netherlands. #Harvard John A. Paulson School of Engineering and Applied Sciences Pierce Hall 29 Oxford Street Cambridge, MA 02138. E-mail: [email protected]; [email protected]

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Abstract Electron surface states in solids are typically confined to the outermost atomic layers and, due to surface disorder, have negligible impact on electronic transport. Here we demonstrate a very different behavior for surface states in graphene. We probe the wavelike character of these states by Fabry-Perot (FP) interferometry and find that, in contrast to theoretical predictions, these states can propagate ballistically over micronscale distances. This is achieved by embedding a graphene resonantor formed by gatedefined p-n junctions within a graphene SNS structure. By combining superconducting Aharanov-Bohm interferometry with Fourier methods, we visualize spatially resolved current flow and image FP resonances due to p-n-p cavity modes. The coherence of the standing wave edge states is revealed by observing a new family of FP resonances which coexist with the bulk resonances. The edge resonances have periodicity distinct from that of the bulk states, manifest in a repeated spatial redistribution of current on and off the FP resonances. This behavior is accompanied by a modulation of the multiple Andreev reflection amplitude on and off resonance, indicating that electrons propagate ballistically in a fully coherent fashion. These results, which were not anticipated by theory, provide a practical route to developing electron analog of optical FP resonators at the graphene edge.

Keywords: ballistic transport; Josephson interferometry; graphene edge states; Fabry-Perot interference; electron optics

Surface states in electronic solids, which occur at the material boundaries, have a wide range of interesting properties. The Tamm-Schockley states, which originate from crystal structure termination or band bending, are localized in the top atomic layer and, being hindered by disorder at the surface, do not contribute current-carrying pathways. 1,2 In contrast, topologically protected surface states in topological insulators or quantum Hall systems allow electrons to propagate coherently without backscattering. 3,4 However, the necessity of topological protection for the coherent propagation of surface states remains an open question. 2


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This work reveals the existence of phase-coherent electron waves at the graphene edges that propagate ballistically over sub-micron length scales. We observe these states, and probe their coherence, by studying standing waves formed at the graphene edges embedded in an electronic Fabry-Perot interferometer. Electron resonators have so far been explored in the momentum domain, in which Fermi momentum is simply tuned with a gate, but little is known about the spatial distribution of current flow due to the challenge of imaging current paths with submicron resolution. To elucidate the role of edge effects on wave interference in graphene, we measure the interference of standing waves in a Josephson junction and image the real space distribution of supercurrent flow using Fraunhofer interferometry. 5 By extracting the spatial structure of current-carrying states in the cavity using Fourier methods, our measurements disentangle edge from bulk current flow and highlight the surprising role of the crystal boundaries on electron wave interference. In a coherent electron cavity, electronic transport is governed by quantum interference of electron plane waves in an enclosed geometry rather than diffusive transport. 6,7 In our system, a pair of p-n junctions serve as electronic mirrors, confining electron waves in analogy to the confinement of light within an optical FP cavity. As an added benefit over optical systems, the Fermi wavelength of electrons in the cavity is electrostatically tunable with a gate, moving the quantized energy levels of the cavity on and off resonance with the Fermi energy of the superconducting leads. The resonances give rise to the wavelengthdependent transmission through the n-p-n cavity resulting in the critical current oscillations. Specifically, in absence of magnetic field, a single electron trajectory with an incidence angle θ and refraction angle θ′ within the cavity yields a transmission probability P = |T (θ, θ′ )|2 , where T (θ, θ′ ) is FP transmission amplitude

T (θ, θ′ ) =

t1 (θ)t2 (θ)eiD , 1 − r1 (θ)r2 (θ)e2iD

3

D = kL cos θ′ ,


(1)

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where t1,2 (θ) and r1,2 (θ) are the angle dependent transmission and reflection amplitudes for two p-n junctions, L is the length of the n-p-n cavity, k is the wavenumber, and D is the one-way optical path. The dependence of the amplitudes t1,2 (θ) and r1,2 (θ) on the angle θ can in principle be obtained from free-electron Hamiltonian describing the p-n junction for a given carriers wavelength. In the presence of magnetic field, using Landau gauge with vector potential A = (−By, 0, 0) the optical path and transmission amplitude are modified by including the Aharonov-Bohm phase:

D→D+

π Φ L tan θ′ 2 Φ0 W

T (θ, θ′ ) → T (θ, θ′ )eiϕAB (y) ,

ϕAB (y) = π

Φ y , Φ0 W

(2)

where we define ϕAB the mean Aharonov-Bohm phase, Φ = BLW is the total magnetic flux through the junction, Φ0 = hc/2e is magnetic flux quantum, and y is the position of the beginning of the trajectory along the leads. Here D is an effective optical path derived as a half-sum of one-way optical paths for the right-moving and left-moving modes in the resonator. This result can be applied to the superconducting transport through the SNS junction mediated by Andreev pairs. For each electron of the pair transmission amplitude through the p-n-p junction is described by by Eq.(1) with the B-dependent contribution to the phase given in Eq.(2). The choice of the model for SNS transport depends on Thouless energy ET = ~/τ , where τ ∼ L/vF is traversal time through the junction. 8 The SNS junctions studied in our experiment are characterized by ET values few times greater than the supeconducting gap ∆. It can be shown that in the limit ET ≫ ∆ a single trajectory with electron transmission probability Eq. (1) gives a contribution to the supercurrent of the form |T (θ, θ′ )|2 sin ϕ 2e∆ p , I(ϕ, θ) = ~ 1 − |T (θ, θ′ )|2 sin2 (ϕ/2)

ϕ = ϕ1 − ϕ2 + 2ϕAB

1 y + L tan θ′ 2

(3)

where the phase difference ϕ includes difference of superconducting phases in the leads ϕ1,2 4


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and the Aharonov-Bohm phase gained by an Andreev pair for the given trajectory (for the derivation of equation (3) see Supporting Information). The resulting interference fringes are dominated by the angles for which both the transmission and reflection are reasonably high (the first harmonic of FP fringes is at its brightest when the product of transmission and reflection coefficients |t(θ)|2 |r(θ)|2 takes a maximum value). In general, the spread of angles for different trajectories in graphene bulk gives rise to a spread of the FP oscillation periods, somewhat reducing the fringe visibility in the net current (see Supplementary Information). In contrast, no suppression is expected for the interference fringes due to edge modes, as discussed in detail later. The experimental manifestation of these standing-wave edge states constitutes a departure from conventional Josephson behavior and arises from the interplay between superconductivity and electron-optics. While the ballistic Josephson effect has been previously probed in one-dimensional systems such as carbon nanotubes and nanowires, the two-dimensional structure of graphene enables use of superconducting Aharonov-Bohm interferometry to understand the contributions of electronic pathways at the edge and in the bulk. In this work, we demonstrate how the presence of an electron resonator inside the Josephson junction can induce novel superconducting phenomena, including unconventional Josephson interference patterns that exhibit lobes as a function of Fermi energy and modulation of the multiple Andreev reflection amplitude depending on interference conditions of electron waves in the cavity. We employ proximity induced superconductivity to shed light on the microscopic nature of electron interference along the crystal boundaries of a graphene resonator. 9–12 Embedding the resonator between a pair of superconducting electrodes is the key step that allows extraction of spatial information on current flow. On a practical level, graphene provides an accessible interface for superconducting electrodes because it is purely a surface material, unlike 2D electron gases embedded in semiconductor heterostructures. Although graphene is not intrinsically superconducting, proximity-induced superconductivity can be mediated by

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phase coherent Andreev reflection at the graphene/superconductor interface. 13,14 This process features an electron-hole conversion by the superconducting pair potential that switches both spin and valley to preserve singlet pairing and zero total momentum of the Cooper pair. 15 In this study, we employ gated mesoscopic Josephson junctions consisting of bilayer graphene suspended between two superconducting Ti/Al electrodes, as well as a graphene device on hBN. The superconductors serve three roles: (1) they create electrostatic potentials that confine electron waves, serving as electronic analogs to mirrors (2) superconducting interferometry can extract spatial information on how current flows through the system, and (3) beyond equilibrium, scattering events between the superconductors and graphene (multiple Andreev reflections) depend critically on resonance conditions and reveal how the resonator couples to the outside world.

A schematic of a suspended graphene Josephson junction is provided in Figure 1a. Accessing a regime where electrons travel ballistically, similar to photons, is indispensable for probing the phase-coherent nature of the edge states. We developed a new method to isolate the flake from charge disorder in the underlying dielectric by suspending it over the back gate electrode, described in the Supplementary Information. This approach combines the high purity of suspended devices with superconductivity and enables the creation of ballistic waveguides where the mean free path le of electrons exceeds channel length L. Similar results are also obtained on a gate-defined resonator in monolayer graphene encapsulated in hBN, discussed later, which enables a higher degree of electronic control over the cavity while preserving sample quality. The superconducting leads serve not only as electronic probes but also induce a resonant electron cavity in the scaling limit le > L (Fig. 1b). 6,7 The graphene in the immediate vicinity of the Ti/Al contact is n-doped by charge transfer , 16 forming an intrinsic n-n or n-p junction near the interface when the graphene has electron or hole carriers, respectively. Taking advantage of contact induced doping to define the resonator, the resulting n-p junctions 6


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the n-p junctions serve as the electronic counterparts to mirrors while the ballistic graphene channel serves as an electron waveguide. The Fermi wavelength of electrons in the cavity, √ expressed as λF = 2π/ πn for a 2D Fermi disk with fourfold degeneracy, is experimentally tunable with a gate voltage Vb that determines the carrier density n. Fabry-Pérot (FP) resonances in ballistic waveguides arise due to reflection from p-n junctions formed near the superconducting leads when the carrier polarity in the graphene region is opposite to the polarity of contact doping. Figure 1b shows a plot of the normal resistance Rn , obtained by sweeping Vb at a fixed bias exceeding the critical current Ic . We observe periodic resistance oscillations at small positive carrier densities (Vb < 0) when n-p-n junction formation is favored and monotonic behavior when doping is unipolar. The dips in Rn originate from constructive interference of electron waves in the resonator and therefore coincide with carrier densities satisfying the condition 2d = mλF , where d is the effective cavity length and m is an integer. The correspondence to FP interference conditions can be seen more clearly in Fig. 1c, which shows that Rn is periodic as a function of 2d/λF . Reproducibility of the oscillation period is demonstrated in three devices of 500 nm length (Fig. 1c), while shorter junctions exhibit larger periods as expected. Quantum confinement between the cavity “mirrors” gives rise to discrete energy levels with spacing hvF /2d, where vF = ~kF /m∗ is the Fermi velocity and and m∗ is the effective electron mass in bilayer graphene. We evaluate this energy scale to be of the order ∼ 0.8 meV using the height of FP diamonds, as measured using voltage bias spectroscopy (Fig. 1d). The interplay between cavity resonances and supercurrent is evident from a resistance colormap as a function of IDC and Vb (Fig. 2a-b) showing critical current oscillations whose period satisfies FP interference conditions, consistent with supercurrent propagation via ballistic charge carriers. 21 As λF in the cavity is tuned with the gate, the quantum levels of the cavity are moved on or off resonance with the Fermi energy of the superconducting √ leads, thus inducing a oscillating critical current periodic in n for bilayer graphene. This phenomena is observed in two independent systems: suspended bilayer graphene resonators

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defined by contact-induced doping (Fig. 2b) and a gate-defined resonator in monolayer graphene on hBN (Fig. 2c and Supplementary Fig. S1), both of which exhibit similar behavior. In total, five suspended bilayer devices are studied with a lithographic distance L between superconducting contacts of 350 to 500 nm and contact width W of 1.5 to 3.2 µm, in addition to one gate-defined monolayer device with cavity dimensions of L = 100 nm and W = 2.7 µm (see Supplementary Information). Figure 2c displays critical current modulations in a gate-defined monolayer resonator whose oscillations are periodic in n, in agreement with a monolayer FP for cavity length ∼ 100 nm.

We employ yet another property of superconductor-normal-superconductor (SNS) systems to gain insight into the coupling between the cavity modes with the superconducting reservoirs. Because the phenomenon of multiple Andreev reflection (MAR) is known to be extremely sensitive to the coupling between electrons in the normal metal and the superconductor, we use voltage bias spectroscopy to map out the interplay between MAR oscillation amplitude and cavity transmission (Fig. 2d-e). The millielectronvolt energy scale associated with FP interference substantially exceeds the Al superconducting gap ∆, allowing one to study the system close to equilibrium conditions for the resonator. Well defined MAR peaks appear at 2∆, ∆, and 2∆/3 when the density is tuned off resonance, while MAR is completely suppressed on resonance, as visible in line cuts of resistance on and off resonance in Fig. 2e (additional data sets are provided in Supplementary Figs. S2, S3-S4). It is notable that the amplitude of the multiple Andreev reflections depends strongly on cavity resonance conditions, thereby providing a direct measure of the tunable coupling between the resonator and the outside world. We hypothesize that the change in visibility of MAR on and off resonance is most naturally explained by changes in the distribution of transmission eigenvalues. When an electron cavity is tuned to the resonant wavelength, a larger fraction of the current is carried by highly transmitting modes than when the cavity is off resonance. Meanwhile, the magnitude 9


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the channel size is ≈ 350 nm, so we expect the short junction limit to qualitatively hold. In order to qualitatively understand the multiple Andreev reflection measurements, we consider a short Josephson junction model. In this limit, the conductance is given by

G(V ) =

Z

1

ρ(T )G(V, T )dT,

(4)

0

with T the transmission probability, ρ the density of transmission eigenvalues, and G(T, V ) the single mode conductance contribution, calculated using expressions from Ref. 22 Because supercurrent is predominately transmitted by bulk modes, as indicated by the Fraunhofer interferometry data (Fig. S6), we simplify our MAR analysis by focusing on resonances of P bulk states. We approximate the integral as a sum, G(V ) ≈ n αn G(Tn , V ), where αn =

Z

Tn +Tn+

ρ(T )dT

(5)

Tn −Tn−

models the contribution of modes having different ranges of transmission probability. Following Refs., 22–24 we compute multiple Andreev reflection curves in the single channel short junction limit, Gn (V ) = G(Tn , V ), for a set of transmission probabilities Tn ∈ {0.3, 0.4, . . . , 1.0}. Comparison with the measured values, Gm (V ), reduces to fitting for the P coefficients αn which minimize the linear system of equations Gm (V )− n αn Gn (V ), for each value of the back gate voltage, Vgate . The theoretical curves have MAR resonances which

are sharper than the experimental ones, so the fitting process is improved by smoothing the single channel results by convolution with a Gaussian function (the width of the Gaussian that we used is 0.03). In order to avoid over-fitting, we limit the number of coefficients αn to 3, corresponding to small T ∈ [0, 0.4), αlow , medium [0.4, 0.8), αmed , and large [0.8, 1] transmission, αhigh . The resulting function has three fitting parameters and reads:

G(V ) = αlow G(T = 0.3, V ) + αmed G(T = 0.6, V ) + αhigh G(T = 0.9, V ).

(6)

Both the amount of smoothing, as well as the value of the superconducting gap to which 11


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and explore the rich interplay between magnetic interference effects and cavity transmission. Upon application of a magnetic field B, a flux Φ penetrates the junction area and induces a superconducting phase difference ∆ϕ(x) = 2πΦx/Φ0 W parallel to the graphene/contact interface, where Φ0 = h/2e is the flux quantum, h is Planck’s constant, and e is the elementary charge. When a flux penetrates the junction area, the critical current Ic (B) exhibits oscillations in magnetic field given by: Z W/2 J(x) · e2πiLBx/Φ0 dx Ic (B) = −W/2

(7)

where L is the distance between superconducting electrodes (Fig. 1). 5,26 This integral expression applies in the wide junction limit, relevant for our system, where L ≪ W and the current density is only a function of one coordinate. Because the critical current Ic (B) equals magnitude of the complex Fourier transform of the real-space supercurrent distribution J(x), the shape of the interference pattern is determined directly by the spatial distribution of supercurrent across the sample. 5,27 To visualize current flow associated with interfering electron waves, we examine the supercurrent modulations in a B field that arise from a Fraunhofer interference. In a conventional Josephson junction with uniform current density, the normalized critical current obeys Ic (B)/Ic (0) = | sin(πΦ/Φ0 )/(πΦ/Φ0 )|. This “single slit” diffraction pattern would be independent of gate voltage, as illustrated in Fig. 3a. In contrast, the fringes in our measured pattern are at variance with this theoretical scenario and exhibit nodes in Ic (B)/Ic (0) as a function of both Vb and B. This is illustrated in Fig. 3b and Supplementary Fig. S6, which reveal the behavior of the normalized interference pattern Ic /Ic (B = 0) at gate voltages corresponding to the on and off resonance conditions (labeled by the red and green dotted lines, respectively). The spatial distribution of supercurrent density J(x), which exhibits gate-dependent modulations along the both edge and bulk of the sample, is provided in Fig. S6c. From this figure, we see the on/off modulation of Andreev resonances occurs mostly

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We hypothesize that separate interferences of boundary states and bulk states are responsible for the experimentally observed critical current fringes. Motivated by the relation between the spatial current distribution J(x) and critical current Ic (B) in Eq. (7), we directly model the spatial distribution of current paths for bilayer graphene in the FP regime (see Supplementary Information). These calculations take into account the guided edge modes that originate from edge potentials, which confine carriers to edge-defined ‘waveguides’ in analogy to the confinement of photons in fiber optics. 28 Energies of the edge modes lie outside the bulk continuum (Fig. 4a), which implies an evanescent-wave decay of the modes into the bulk. The resulting modes are quasi one-dimensional, propagating ballistically as plane waves along the graphene edges. Applying the FP quantization condition in the p-n-p region leads to a sequence of FP maxima positioned at kn = πn/L, where n is an integer and L represents the distance between superconducting contacts. Since the FP resonances for the edge modes originate from head-on transmission and reflection, the corresponding FP fringes are much stronger than the angle-averaged FP fringes for the bulk states (Supplementary Fig. S9). As shown in the theoretical dispersion in Fig. 4a, the interference conditions at the edge and in the bulk should not coincide due to the difference in the carrier dispersion at the edge and in the bulk, as well as due to the angle-dependence of the FP period for bulk carriers. Hence a gradual increase of doping will trigger repeated switching between the bulk-dominated and edge-dominated regimes, with the current distribution switching from spatially uniform to edge like, accordingly. Qualitatively, this would be manifested in the dependence of measured critical current on applied magnetic field as switching between Fraunhofer and more SQUID-like behavior (Fig. 4b-c). In this paper, the term “edge-dominated” refers specifically to Fraunhofer interference patterns that feature an enhancement in the critical currents of the side lobes (outside the B=0 central lobe) when compared to the standard 1/B envelope expected for a uniform supercurrent distribution (see Fig. 4b). To relate this scheme with the transport data, we model FP resonances in accordance

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with the approach described in Ref. 28 Assuming that the edge potential is sufficiently shortranged, we approximate it with a delta function and obtain the density of persistent current along the edge (see Supplementary Information). To translate this prediction into an experimentally observable quantity, we model the superconducting interference pattern Ic (B) using the energy dispersion relation of edge modes (Fig. 4a). This result, plotted in Fig. 4b, captures the key features of the raw data in Fig. 4c, namely the redistribution of current on and off resonance as well as the suppression of the side lobes’ intensity on resonance (Supplementary Fig. S8). Thus, the measurements are consistent with a model that features separate FP interference of guided-wave edge currents in parallel to interference of bulk modes. This further suggests that the quasi-1D edge currents previously observed have ballistic character. Despite the model’s simplified nature, which neglects disorder and finite temperature effects, this scheme captures the essential features of the measurements. While the edge potential featured in this simulation accommodates a single edge channel, we note that the number of guided modes may exceed one for stronger potentials. While the edge potential featured in this simulation accommodates a single edge channel, we note that the number of guided modes may exceed one for stronger potentials. In this case, each mode would contribute independently to the interference pattern. In summary, we utilize different aspects of proximity-induced superconductivity, particularly Fraunhofer interferometry and Andreev scattering, as new tools to resolve optics-like phenomena associated with electron waves confined within a ballistic graphene Josephson junction. This enables real-space visualization of cavity modes in a graphene FP resonator, which reveals surprising redistribution of current on and off resonance and provides direct evidence of the ballistic nature of guided edge currents. These results constitute a strong departure from conventional Josephson behavior in graphene and motivate further exploration of new effects at the intersection of superconductivity and optics-like phenomena. The ballistic character and coherent nature of edge mode transport in a non-topological material such as graphene, is a puzzling behavior that calls for novel theoretical ideas going

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beyond the usual scenarios for topological protection. New theory must explain why ballistic propagation at exceptionally long length scales occurs despite strong intrinsic disorder present at the edge. One possible scenario, the so-called weak guiding mechanism, has been outlined in this work. Ballistic transport at a strongly disordered graphene edge has been entirely overlooked by previous work, both experimental and theoretical. As such, establishing the presence of a new conduction pathway will dramatically boost our understanding of charge transport mechanisms in atomically thin conductors.

Conflict of Interest Disclosure The authors declare no competing financial interest.

Acknowledgement The authors thank O. Dial, B. Halperin, V. Manucharyan, and J. Sau for helpful discussions. This work is supported by the Center for Integrated Quantum Materials (CIQM) under NSF award 1231319 (LSL and OS) and the U.S. DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0001819 (PJH, MTA, AY). Nanofabrication was performed at the Harvard Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN) supported by NSF award ECS-0335765. AA was supported by the Foundation for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO/OCW). ICF was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Project MUNATOP, the US-Israel Binational Science Foundation, and the Minerva Foundation for support.

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Supporting Information Available This information is available free of charge via the Internet at http://pubs.acs.org • JosephsonFabryPerotsupplement.pdf: Supplementary Information. 15 pages, including 9 supplementary figures.

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